Considering the case of a pure sinusoidal tone, e.g. the tuning A note at $440 \text{Hz}$, how can one mathematically represent the pressure wave resulting? For the sake of simplicity, I want to ignore any attenuation.
If I use $A(t) = \sin(2 \pi f t)$, $t$ being time and $A$ amplitude, I guess that means how the amplitude (or pressure, in an unspecified unit) varies with time, but such a function can only refer to a single point, can't it?
Given that sound waves propagate in 3D space, how can I represent that in this very simple case of a pure sinusoidal A tone? My understanding is amplitude varies as a sin function with time in every point of space and also between different points, again as a sin function.
So I am looking for a 3D representation, something like $A(x,y,z,t)$, but I am also interested in restricting it to a certain direction and find an $A(x,t)$, which should be easy because of symmetry.
I wonder if using $A(t) = \sin(2 \pi f t)$ — how amplitude varies with time in a single point in space — would be enough to demonstrate what happens when I sum up two pure sounds at different frequencies (obtaining the so called "beats"), but I am still curious about full mathematical representation of this kind of longitudinal waves.
The typical representation of a traveling plane sine wave would be something like $$ f(x,t) = A \sin(\omega t - kx) $$ where $\omega = 2\pi f $ is the angular frequency and $k$ is the wave number, related to the wavelength $\lambda$ by $\lambda = \frac{2\pi} k$, or to the speed $v$ of the wave by $k = \omega/v$.